This paper extends foliation concepts to singular foliations using Lie ∞-algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie ∞-algebroids to define modular class. result Geometric meaning of modular class as an obstruction to universal Lie ∞-algebroids. This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic funct…
Homotopy theory applied to singular foliations leads to new results.
problem Existence and uniqueness of universal L∞-algebroids for singular foliations. method Applied homotopy theory to left semi-model categories and L∞-algebroids. result Recovery of results similar to Laurent-Gengoux and al. about universal L∞-algebroids. New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
A singular (or Hermann) foliation on a smooth manifold M can be seen as a subsheaf of the sheaf X of vector fields on M. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…
Holonomy defined for singular leaves in foliations.
problem Defining holonomy for singular structures in foliations.
method Introducing a sequence of group morphisms from π_n(L) to π_{n-1} of the universal Lie ∞-algebroid.
result Holonomy sequence relates to Androulidakis-Skandalis and Brahic-Zhu constructions.
Computes L∞-algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes L∞-algebroid structure. result Provides invariants and constant-rank replacements of singular foliations.
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. This paper studies Hopf algebras from dg manifolds.
problem Understanding Hopf algebras from the perspective of dg manifolds.
method Analyzes the universal enveloping algebra of Lie algebra objects in homotopy categories of dg modules.
result The universal enveloping algebra of the Lie algebra object is a Hopf algebra.
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
The paper constructs L∞-algebras from contact Courant algebroids and isotropic subbundles.
problem Understanding the structure of contact Courant algebroids and their associated L∞-algebras. method The construction of L∞-algebras from L-Courant algebroids and isotropic subbundles. result A relationship between constructed L∞-algebras is established by a morphism. In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid L-and the natural generalization to dg Lie algebroids-provides an (essentially unique) L∞ space. More precisely, we construct a faithful functor from the category of Lie algebroids …
Study modular class of Lie ∞-algebroids and their adjoint actions.
problem Understanding the modular class and adjoint actions of Lie ∞-algebroids.
method Equivalence of descriptions, homotopy invariance, explicit actions and dualities.
result Homotopy invariance of modular classes and explicit adjoint actions.
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical L∞-morphisms for higher Courant algebroids. method Develop a general framework for arbitrary r. result Affirmative answer to Zambon's question for higher degrees.
Constructs brane current algebras from QP-manifolds, generalizing string currents.
problem Constructing brane current algebras from QP-manifolds.
method Using Poisson algebra and QP-manifolds (symplectic L∞-algebroids), the paper derives a universal geometric form for Poisson brackets of brane currents. result Derives a universal expression for 't Hooft anomaly in the presence of fluxes.
We show that L∞-algebroids, understood in terms of Q-manifolds can be described in terms of certain higher Schouten and Poisson structures on graded (super)manifolds. This generalises known constructions for Lie (super)algebras and Lie algebroids.
Geometrically deforms L∞ algebras to Lie algebroids, revealing new invariants.
problem Classifying geometric invariants of L∞ algebras arising from vector bundles. method Define geometric deformations of curved L∞ algebras and show they correspond to Lie algebroid structures. result Geometric deformations of L∞ algebras classify new geometric invariants. Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. Split Courant algebroids linked to special algebra structures.
problem Understanding the structure of split Courant algebroids.
method Established a correspondence with multiplicative curved L∞-algebras. result Split Courant algebroids correspond to multiplicative curved L∞-algebras. Curved L∞ spaces form a category of fibrant objects.
problem Understanding the structure of curved L∞ spaces. method Proving L∞ spaces over dg manifolds form a category of fibrant objects. result Transitive L∞ algebroids over dg manifolds also form a category of fibrant objects. Shifted symplectic Lie and L∞ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetri…
We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable L∞-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geomet…
The L∞-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one L∞-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…
We study Maurer-Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term $L_\infty…
Universal construction for Lie algebroids from anchored bundles with connections.
problem Creating a universal framework for Lie algebroids from anchored bundles with connections.
method Adapting Kapranov's construction to anchored bundles with arbitrary connections, showing compatibility with Lie algebroid structure.
result Universal connection ildeabla on FR(E) compatible with Lie algebroid structure, turning (FR(E),ildeabla) into a Cartan-Lie algebroid. The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. Constructs a Lie groupoid integrating singular foliations.
problem Integrating singular foliations into higher Lie groupoids.
method Recursive use of bi-submersions and geometric resolutions.
result Finite-dimensional Lie groupoid integrating singular foliations.
We introduce the notion of left (and right) quasi-Loday algebroids and a "universal space" for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.
New L∞ algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an L∞ algebra is associated with each Dirac-Jacobi structure. result There is a one-to-one correspondence between MC elements of the L∞ algebra and small deformations of the Dirac-Jacobi structure. The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of L∞-algebras for deformations. result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.
New algebraic structure derived from Kähler manifolds.
problem Understanding algebraic structures on differential forms.
method Introducing L∞[1] R-algebras and proving linearization theorems. result Induced L∞[1] R-algebra structures on Γ(L) are linearizable under certain conditions. Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new L∞-algebras of higher symmetries. Deformations of a Courant Algebroid E and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric is fixed. In this paper, we attack the same problem in a setting that allows the pseudo-Euclidean metric to deform. Thanks to Roytenberg, a Courant algebroid is equivalent to…
The paper resolves singular foliations through a series of blowups.
problem Singular foliations that cannot be resolved directly.
method Applying Nash modifications to the universal Lie ∞-algebroid of a singular foliation.
result Any singular foliation becomes a Debord foliation after one blowup.
Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to L∞-algebroids to show th…
In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac struct…
Lie-Rinehart algebras over C∞-rings defined and studied.
problem Defining and studying Lie-Rinehart algebras over C∞-rings. method Defining Lie-Rinehart algebras over C∞-rings and showing their relationship with Poisson C∞-rings. result A natural Poisson bracket on the C∞-ring associated with a Lie-Rinehart algebra over a C∞-ring. Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
problem Cohomology of Lie algebroids over algebraic spaces.
method Express hypercohomology as a derived functor, simplify via Čech cohomology, define Hochschild hypercohomology, present Hochschild-Kostant-Rosenberg theorem.
result Presented a version of Hochschild-Kostant-Rosenberg theorem for locally free Lie algebroids.
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,∇) where B ⊂ A are regular Lie algebroids, both over the same regular foliated manifold (M,…
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
The paper extends Chern-Weil-Lecomte map to L∞-algebras.
problem Defining characteristic classes for L∞-algebra extensions. method Using the Chern-Weil-Lecomte map to define characteristic classes in an L∞-algebra setting. result Unified definition of several known cohomology classes.
This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebroïd structure. This structure is a Courant algebroïd if and only if the the curvature κ of the Cartan connection vanishes. In all other cases, if the parabolic geometry is regular, there does not exist a natural univ…
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale L…
New Lie groupoid and algebroid constructed for octonionic Hopf foliation.
problem No known Lie group action generates the singular octonionic Hopf foliation.
method Constructs a G2-equivariant Lie groupoid and Lie algebroid.
result Minimal Lie algebroid and groupoid generate the singular octonionic Hopf foliation.
We investigate Nijenhuis deformations of L∞-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie n-algebras and n-plectic manifolds, are included.
We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal…